Unramified extensions October 20, 2009Posted by Akhil Mathew in algebra, algebraic number theory, number theory.
Tags: discrete valuation rings, Nakayama's lemma, unramified extensions
As is likely the case with many math bloggers, I’ve been looking quite a bit at MO and haven’t updated on some of the previous series in a while.
Back to ANT. Today, we tackle the case . We work in the local case where all our DVRs are complete, and all our residue fields are perfect (e.g. finite) (EDIT: I don’t think this works out in the non-local case). I’ll just state these assumptions at the outset. Then, unramified extensions can be described fairly explicitly.
So fix DVRs with quotient fields and residue fields . Recall that since , unramifiedness is equivalent to , i.e.
Now by the primitive element theorem (recall we assumed perfection of ), we can write for some . The goal is to lift to a generator of over . Well, there is a polynomial with ; we can choose irreducible and thus of degree . Lift to and to ; then of course in general, but if is the maximal ideal in , say lying over . So, we use Hensel’s lemma to find reducing to with —indeed is a unit by separability of .
I claim that . Indeed, let ; this is an -submodule of , and
because of the fact that is generated by as a field over . Now Nakayama’s lemma implies that .
Proposition 1 Notation as above, if is unramified, then we can write for some with ; the irreducible monic polynomial satisfied by remains irreducible upon reduction to .
There is a converse as well:
Proposition 2 If for whose monic irreducible remains irreducible upon reduction to , then is unramified, and .
Consider . I claim that . First, is a DVR. Now is a finitely generated -module, so any maximal ideal of must contain by the same Nakayama-type argument. In particular, a maximal ideal of can be obtained as an inverse image of a maximal ideal in
by right-exactness of the tensor product. But this is a field by the assumptions, so is the only maximal ideal of . This is principal so is a DVR and thus must be the integral closure , since the field of fractions of is .
Now , so unramifiedness follows.
Next up: totally ramified extensions, differents, and discriminants.